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Jan
23
accepted Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Jan
23
awarded  Scholar
Jan
23
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Jan
23
accepted Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Jan
23
comment Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Many thanks! I should have checked before I stated there were only three important constants. What I was really hoping for with this question was to see if some kind of deep geometric intuition exists -- e.g. perhaps the circle, the relationship via e between differentiation and integration, and i might (I was thinking) essentially encapsulate a huge amount of geometric information about the relationships between the dimensions, and therefore maybe that's why they show up all the time in formulas. But perhaps it's not so, as you have pointed to many examples of other important constants.
Jan
21
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Alex Best, thanks for your link re Hamel Basis. It's really interesting, this Axiom of Choice -- it seems to relate directly to what was bothering me.
Jan
21
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Michael E2, thank you for your point about the surreal numbers. I will take a look. They are a lot to absorb and I don't know if, in effect, they do the same thing in constructing the surreals that originally bugged me about the reals, which was, this whole "equivalence class of Cauchy sequences" or Dedekind cut construction. In which case, I could have the same potential issue that motivated my question in the first place (but which may just require further thought). I do see an equivalence class in the definition of the surreals on Wikipedia, but I have not had time to really study them.
Jan
21
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Hurkl, to my knowledge, what you have proposed doesn't work. If you can cite any specific authority (link to a book or paper), that would help me understand what you are trying to say.
Jan
19
comment Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Asaf, re math not being about numbers anymore, I am not sure I understand that statement completely. Don't all the letters ultimately represent sets of numbers, in the end? On the second point, I am sure I am wrong about calling anything "esoteric", maybe a bad choice of words. So I withdraw that adjective. It is just really interesting to me that so few numbers are so important and I wonder if anyone has a perspective on why that is the case.
Jan
19
awarded  Editor
Jan
19
revised Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
added 50 characters in body
Jan
19
asked Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
Jan
19
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
In other words, I am saying that, in your terms, as far as I can tell so far: "B does not suffice instead of A" because B requires A in order to exist -- B (the hyperreals or nonstandard R) apparently is defined as an extension of A (the reals) at least in its axiomatic definition that Keisler uses in his calculus text using nonstandard R. You may know more about the constructive definition of nonstandard R that may not, for all I know, assume R, unlike the axiomatic definition I read.
Jan
19
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Michael E2, my point was just that nonstandard R is built on standard R - it needs standard R, whereas standard R does not need nonstandard R. I base this statement on a limited amount of reading on the subject including the calculus book which relies on the transfer principle which I would summarize in essence as saying "assume R already exists and all its axioms are true, then we add infinitesimals and hyperreals to do calculus". To be fair, this is the "axiomatic" way to define nonstandard R (as it says on Wikipedia) - perhaps their "constructive" definition does not cheat by using R.
Jan
19
awarded  Commentator
Jan
19
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
I will post back if/when after further thinking, I feel I really honestly "get" why this Cauchy/Dedekind thing is such an absolutely necessity. I do see how there is this need for the "continuum" to be proven to "exist" so that there aren't any "holes", and I sort of see how it is helpful to actually be able to say "please refer to exhibit A, your Honor: equivalence classes of Cauchy sequences! These are the same thing as this ordered continuum that we are trying to prove exists, so that means that this ordered continuum thing exists." All these comments have been helpful.
Jan
18
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Many thanks for your answer. Obviously still thinking this through and all these responses really help.
Jan
18
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Something is still bugging me about all this and that is, aren't e and Pi counterexamples! Pi is not an equivalence class of Cauchy sequences. It is not a Dedekind cut. It is a real number. Obviously, it is intrinsic to the relationship between the length of the radius and the area (and circumference) of the circle. So: can real numbers be fully classified by their geometric relationships or implications, and if so, do those geometric relationships or implications form a basis for describing the reals that is better than using the mere property "every real number gets converged to"?
Jan
18
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
These nonstandard analysis guys I think actually explain why somebody can't do a "nonstandard analysis" version of the reals that assumes only the rationals and augments them with numbers "infinitely close" to the rationals. Their reason is that if you pick any real number, it is demonstrably not "infinitely close" to any particular rational number whatsoever. In other words, that idea doesn't work. But this whole thing with R still really bugs me; I just really dislike this "infinite sequences (or infinite sets in the case of Dedekind) of numbers" definition of R.
Jan
18
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
If they were really "constructing the nonstandard reals" then they would say something like "assume the rationals" and add to them, numbers infinitesimally close to the rationals. They do NOT do this! Instead, they say "assume the reals" (!) for example citing the free text Elementary Calculus: an Infinitesimal Approach by Keisler online at math.wisc.edu/~keisler/calc.html page 27: "The real numbers form a subset of the hyperreal numbers..." They use the reals to define the hyperreal extension of R. Perhaps an awesome way to do calculus but so far not helpful to me re R.